The Stacks project

Lemma 102.11.5. Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be quasi-compact and quasi-separated morphisms of algebraic stacks. Let $\mathcal{F}$ be a quasi-coherent sheaf on $\mathcal{X}$. Then there exists a spectral sequence with $E_2$-page

\[ E_2^{p, q} = R^ pg_{\mathit{QCoh}, *}(R^ qf_{\mathit{QCoh}, *}\mathcal{F}) \]

converging to $R^{p + q}(g \circ f)_{\mathit{QCoh}, *}\mathcal{F}$.

Proof. By Cohomology on Sites, Lemma 21.14.7 the Leray spectral sequence with

\[ E_2^{p, q} = R^ pg_*(R^ qf_*\mathcal{F}) \]

converges to $R^{p + q}(g \circ f)_*\mathcal{F}$. By the results of Proposition 102.8.1 all the terms of this spectral sequence are objects of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Z})$. Applying the exact functor $Q_\mathcal {Z} : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Z}) \to \mathit{QCoh}(\mathcal{O}_\mathcal {Z})$ we obtain a spectral sequence in $\mathit{QCoh}(\mathcal{O}_\mathcal {Z})$ covering to $R^{p + q}(g \circ f)_{\mathit{QCoh}, *}\mathcal{F}$. Hence the result follows if we can show that

\[ Q_\mathcal {Z}(R^ pg_*(R^ qf_*\mathcal{F})) = Q_\mathcal {Z}(R^ pg_*(Q_\mathcal {X}(R^ qf_*\mathcal{F})) \]

This follows from the fact that the kernel and cokernel of the map

\[ Q_\mathcal {X}(R^ qf_*\mathcal{F}) \longrightarrow R^ qf_*\mathcal{F} \]

are parasitic (Lemma 102.10.2) and that $R^ pg_*$ transforms parasitic modules into parasitic modules (Lemma 102.9.3). $\square$

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